The exemplary embodiment relates to a method and system of selling goods and services. It finds particular application in connection with sales by lottery and in some cases, for establishing optimal pricing for later sales by conventional methods.
A buyer considering a purchase is likely to buy an item if the value that he places on the item is greater than the price he will have to pay for the item. However, buyers rarely value only one alternative when considering a purchase. There may be near-substitute products (e.g., multiple TVs), alternative payment methods (e.g., corresponding to different buyer time discounting or interest rates), varying quantities and qualities of the same product, or different expectations about mean future values (e.g., when advance-selling the same item, which can result in arbitrarily more profit than selling one item at a time). All of these factors can affect whether or not a buyer will purchase a given item. From the seller's perspective, each potential buyer has his own value placed on an item, which the seller does not know and can only estimate from prior sales (if a prior buyer has paid the price, the buyer's value must have been higher). Sellers generally want to maximize their profit, or the welfare of buyers without making a loss, or some combination thereof, which are functions of the sales price and the number of goods sold. The seller may adjust the price over time to obtain a better idea of what buyers are willing to pay and how much surplus buyers are making, but this process can be time consuming and expensive for the seller. Frequent manipulation of pricing can also be problematic, since it can annoy buyers. If the price is set too high then buyers may go elsewhere, and may not return to the seller even if the seller later lowers the price.
For purposes of the embodiments described herein, a buyer is described by a point in value space at a particular time, where each axis corresponds to the valuation for one alternative item. For example, where two different items are being sold, each item has its own value axis and the value space is two dimensional. A seller's belief about buyers can be considered as a probability density over the value space. The seller's objective is to divide the value space into regions that are served by different contracts. These regions are known as (market) segments.
A lottery generally involves a distribution of tokens, such as tickets or virtual tokens, for a given price, with the understanding that one or more of the tokens, drawn randomly from the set of tokens, entitles the holder to a given item or group of items. Lotteries have been widely used as a game of chance in which the ticket holder gains satisfaction from the remote possibility of winning a large prize, even though the probability of winning generally does not warrant the cost of the ticket.
Lotteries have not been widely used for selling goods and services, however, where the item is traditionally sold for a fixed price. Moreover, it has been shown that in the case of lotteries for a single item, fixed price sales are generally equally effective. This is because when buyer valuations are one-dimensional, a seller that knows the probability density of buyer valuations would never find it better to provide contracts that involve a lottery. It was widely-believed until recently that this was also the case for multidimensional valuations, where a buyer is placing a valuation on two or more items. However, more recently, it has been proposed that lotteries do form part of the optimal mechanism for multidimensional valuations (see, Thanassoulis, J., Haggling over substitutes. J. Economic Theory, 117:217-245 (2004)).
Optimal multidimensional mechanisms have been characterized in detail via a function that describes a buyer's utility when they purchase the best possible contract for themselves. This function is known as the mechanism function (see, Manelli, A., Vincent D., Multidimensional mechanism design: Revenue maximization and the multiple-good monopoly. J. Economic Theory, 137:153-185 (2007)). Briest, et al. investigated the profit from an optimal lottery relative to the profit from an optimal non-lottery pricing scheme. They showed that the gain is three in two dimensions, unbounded in four and higher dimensions (Briest, P., Chawla, S., Kleinberg, R., Weinberg, S., Pricing Randomized Allocations. Proc. 21st Annual ACM-SIAM Symp. on Discrete Algorithms, Ed., Moses Charikar (January 2010)). Surprisingly, optimal lotteries may be found efficiently using linear programming or semi-definite programming. Effective methods for solving such problems are discussed by Aguilera and Morin (Aguilera, N., Morin, P., On convex functions and the finite element method. SIAM Journal on Numerical Analysis 47(4):3139-3157, (2009)).
One problem with lotteries is that the value that a buyer places on a lottery is not only influenced by the valuations that the buyer places on the items, but also by the extent to which the buyer is risk averse. Thus, lotteries may not perform optimally according to the predictions of classical economics.